Mathematics Commons^™

Mathematics At Chartres Cathedral, Richard Stout

ACMS Journal 2004

Chartres Cathedral is highly regarded as a magnificent Gothic structure including beautiful stained glass windows and striking sculptures. Mathematics, especially geometry, played a central role in the design of this cathedral. This paper explores that mathematics and how it contributes to the overall effectiveness of the design.

Mathematics And Values: Can Philosophy Guide Projects?, Michael H. Veatch

ACMS Journal 2004

The philosophy of mathematics has provided insight on questions of foundations and mathematical truth; however, it has not been very fruitful in guiding the practice of mathematics. This paper attempts to find points of contact between a Christian worldview and the choice of mathematical projects and methods. Three areas are considered: (i) dubitability in current research, (ii) the intrinsic value of contemporary mathematics to contemporary society, and (iii) the affirmation of human value in the use of mathematics. Finally, a framework for valuing mathematics is proposed as an encouragement to think more deeply about how a Christian might choose a …

Travelling Fronts In Cylinders And Their Stability, Jerrold W. Bebernes, Comgming Li, Yi Li

Mathematics and Statistics Faculty Publications

No abstract provided.

The Nordstrom–Robinson Code Is Algebraic-Geometric, Judy L. Walker

Department of Mathematics: Faculty Publications

The techniques of algebraic geometry have been widely and successfully applied to the study of linear codes over finite field since the early 1980’s. Recently, there has been an increased interest in the study of linear codes over finite rings. In a previous paper [10], we combined these two approaches to coding theory by introducing and studying algebraic-geometric codes over rings. In this correspondence, we show that the Nordstrom–Robinson code is the image under the Gray mapping of an algebraic geometric code over Z = 4Z.

A Dual Approach To Constrained Interpolation From A Convex Subset Of Hilbert Space, Frank Deutsch, Wu Li, Joseph D. Ward

Mathematics & Statistics Faculty Publications

Many interesting and important problems of best approximationare included in (or can be reduced to) one of the followingtype: in a Hilbert spaceX, find the best approximationPK(x) to anyx∈Xfrom the setK≔C∩A−1(b),whereCis a closed convex subset ofX,Ais a bounded linearoperator fromXinto a finite-dimensional Hilbert spaceY, andb∈Y. The main point of this paper is to show thatPK(x)isidenticaltoPC(x+A*y …

Positive Solution Curves Of Semipositone Problems With Concave Nonlinearities, Alfonso Castro, Sudhasree Gadam, Ratnasingham Shivaji

All HMC Faculty Publications and Research

We consider the positive solutions to the semilinear equation:

-Δu(x) = λf(u(x)) for x ∈ Ω

u(x) = 0 for x ∈ ∂Ω

where Ω denotes a smooth bounded region in R^N (N > 1) and λ > 0. Here f :[0, ∞)→R is assumed to be monotonically increasing, concave and such that f(0) < 0 (semipositone). Assuming that f'(∞) ≡ lim t→∞ f'(t) > 0, we establish the stability and uniqueness of large positive solutions in terms of (f(t)/t)'. When Ω is a ball, we determine the exact number of positive solutions for each λ > 0. We also obtain the geometry of the branches of positive solutions completely and establish how …

Global Attractors From The Explosion Of Singular Cycles, Carlos Arnoldo Morales, Maria José Pacífico, Enrique Ramiro Pujals

Publications and Research

Abstract:

In this paper we announce recent results on the existence and bifurcations of hyperbolic systems leading to non-hyperbolic global attractors.

Résumé:

Nous présentons dans cette Note des résultats récents concernant l’existence et les bifurcations d’un nouvel attracteur global chaotique.

The Duals Of Warfield Groups, Peter Loth

Mathematics Faculty Publications

A Warfield group is a direct summand of a simply presented abelian group. In this paper, we describe the Pontrjagin dual groups of Warfield groups, both for the p-local and the general case. A variety of characterizations of these dual groups is obtained. In addition, numerical invariants are given that distinguish between two such groups which are not topologically isomorphic.

Eigenfunction And Harmonic Function Estimates In Domains With Horns And Cusps, Michael Cranston, Yi Li

Mathematics and Statistics Faculty Publications

No abstract provided.

An Invariance Property Of Common Statistical Tests, N. Rao Chaganty, A. K. Vaish

Mathematics & Statistics Faculty Publications

Let A be a symmetric matrix and B be a nonnegative definite (nnd) matrix. We obtain a characterization of the class of nnd solutions Σ for the matrix equation AΣA = B. We then use the characterization to obtain all possible covariance structures under which the distributions of many common test statistics remain invariant, that is, the distributions remain the same except for a scale factor. Applications include a complete characterization of covariance structures such that the chisquaredness and independence of quadratic forms in ANOVA problems is preserved. The basic matrix theoretic theorem itself is useful in other characterizing …

Scattering From Stellar Acoustic-Gravity Potentials: Ii. Phase Shifts Via The First Born Approximation, J. A. Adam, I. Mckaig

Mathematics & Statistics Faculty Publications

Using the first Born approximation, properties of the scattering phase shift are investigated for waves that are scattered by a schematic representation of a large-scale “stellar potential,” i.e., one for which the star itself is viewed as the potential inducing a phase shift in an incoming wave. In particular, the phase shift properties are examined as functions of the relative wavenumber (α) and the azimuthal wavenumber (l), high l-values being of interest in helioseismology.

Convolution And Fourier-Feynman Transforms, Chull Park, David Skough

Department of Mathematics: Faculty Publications

In this paper, for a class of funtionals on Wiener space of the form F(x) = exp{∫^T₀ f(t, x(t)) dt}, we show that the Fourier-Feynman transform of the convolution product is a product of Fourier-Feynman transforms. This allows us to compute the transform of the convolution product without computing the convolution product.

Collected Papers, Vol. 2, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Existence And Bifurcation Of The Positive Solutions For A Semilinear Equation With Critical Exponent, Yinbin Deng, Yi Li

Mathematics and Statistics Faculty Publications

In this paper, we consider the semilinear elliptic equation[formula]Forp=2N/(N−2), we show that there exists a positive constantμ*>0 such that (∗)_μpossesses at least one solution ifμ∈(0, μ*) and no solutions ifμ>μ*. Furthermore, (∗)_μpossesses a unique solution whenμ=μ*, and at least two solutions whenμ∈(0, μ*) and 2<NN⩾6, under some monotonicity conditions onf((1.6)) we show that there exist two constants 0<μ_**⩽μ**<μ* such that problem (∗)_{μ …}

Self-Consistency: A Fundamental Concept In Statistics, Thaddeus Tarpey, Bernard Flury

Mathematics and Statistics Faculty Publications

The term ''self-consistency'' was introduced in 1989 by Hastie and Stuetzle to describe the property that each point on a smooth curve or surface is the mean of all points that project orthogonally onto it. We generalize this concept to self-consistent random vectors: a random vector Y is self-consistent for X if E[X|Y] = Y almost surely. This allows us to construct a unified theoretical basis for principal components, principal curves and surfaces, principal points, principal variables, principal modes of variation and other statistical methods. We provide some general results on self-consistent random variables, give …

Positive Solutions For A Semilinear Elliptic Problem With Critical Exponent, Ismail Ali, Alfonso Castro

All HMC Faculty Publications and Research

No abstract provided in article.

The Exponential Stability Of A Coupled Hyperbolic/Parabolic System Arising In Structural Acoustics, George Avalos

Department of Mathematics: Faculty Publications

We show here the uniform stabilization of a coupled system of hyperbolic and parabolic PDE’s which describes a particular fluid/structure interaction system. This system has the wave equation, which is satisfied on the interior of a bounded domain Ω, coupled to a “parabolic–like” beam equation holding on ∂Ω, and wherein the coupling is accomplished through velocity terms on the boundary. Our result is an analog of a recent result by Lasiecka and Triggiani which shows the exponential stability of the wave equation via Neumann feedback control, and like that work, depends upon a trace regularity estimate for solutions of hyperbolic …

Droplet Evaporation And Deformations In An Amplitude Modulated Ultrasonic Field, Nihad E. Daidzic, Rene Stadler, Adrian Melling

Aviation Department Publications

The aim of the report presented is the measurements of droplet oscillations.

Bounds On Squares Of Two-Sets, Daniel Slilaty, Jeffrey Vanderkam

Mathematics and Statistics Faculty Publications

Let G be a finite group and let pi(G) denote the proportion of (x, y) ∈ G2 for which the set {x2,xy,yx,y2} has cardinality i. We show that either 0 < pi(G) + p2(G) ≤ 1/2 or pi(G) + p2(G) = 1, and that either p4(G) = 0 or 5/32 ≤ p4(G) ≤ 1. Each of the preceding inequalities are the best possible.

Stability Analysis Of A Model For The Defect Structure Of Yba2cu3ox, Gregory Kozlowski, Tom Svobodny

Physics Faculty Publications

Unusual microstructures of YBa₂Cu₃O_x (123) crystals have been observed. These structures have been shown to pass very high transport currents. A model of the solidification of 123 from a melt with Y₂BaCuO₅ (211) inclusions indicates that the stability of the 123 interface can depend on the sizes of the 211 inclusions. The observed formations are interpreted in the light of this instability.

Data Compression Based On The Cubic B-Spline Wavelet With Uniform Two-Scale Relation, S. K. Yang, C. H. Cooke

Mathematics & Statistics Faculty Publications

The aim of this paper is to investigate the potential artificial compression which can be achieved using an interval multiresolution analysis based on a semiorthogonal cubic B-spline wavelet. The Chui-Quak [1] spline multiresolution analysis for the finite interval has been modified [2] so as to be characterized by natural spline projection and uniform two-scale relation. Strengths and weaknesses of the semiorthogonal wavelet as regards artificial compression and data smoothing by the method of thresholding wavelet coefficients are indicated.

A Center-Unstable Manifold Theorem For Parametrically Excited Surface Waves, Larry Turyn

Mathematics and Statistics Faculty Publications

When fluid in a rectangular tank sits upon a platform which is oscillating with sufficient amplitude, surface waves appear in the ''Faraday resonance.'' Scientists and engineers have done bifurcation analyses which assume that there is a center manifold theory using a finite number of excited spatial modes. We establish such a center manifold theorem for Xiao-Biao Lin's model in which potential flow is assumed but an artificial dissipation term is included in the system of partial differential equations on the free surface. We use interpolation spaces developed by da Prate and Grisvard, establish maximal regularity for a family of evolution …

Exact Multiplicity Results For Boundary Value Problems With Nonlinearities Generalizing Cubic, Philip Korman, Yi Li, Tiancheng Ouyang

Mathematics and Statistics Faculty Publications

No abstract provided.

The Support Of Measure-Valued Branching Processes In A Random Environment, Donald A. Dawson, Yi Li, Carl Mueller

Mathematics and Statistics Faculty Publications

We consider the one-dimensional catalytic branching process intro­duced by Dawson and Fleischmann, which is a modification of the super­ Brownian motion. The catalysts are given by a nonnegative infinitely divisible random measure with independent increments. We give sufficient conditions for the global support of the process to be compact, and sufficient conditions for noncompact global support. Since the catalytic process is related to the heat equation, compact support may be surprising. On the other hand, the super-Brownian motion has compact global support. We find that all nonnegative stable random measures lead to compact global support, and we give an example …

Stability Of Growing Front Of Yba(2)Cu(3)O(X) Superconductor In The Presence Of Pt And Ceo(2) Additions, Gregory Kozlowski, Tom Svobodny

Physics Faculty Publications

Distinctive microstructures of textured YBa₂Cu₃O_x (123) superconductors were examined by scanning electron microscopy and metallurgical microscopy. The samples were synthesized under a residual thermal gradient by using a modified melt textured growth on a Y₂BaCuO₅ (211) substrate. Also, the unidirectional solidification by a zone‐melting method was performed to fabricate 123 superconducting bars up to 12 cm long placed on the 211 substrate in the horizontal arrangement, with a growth rate R=0.5 mm/h and a temperature gradient of G=20 °C/cm (G/R=400 °C h/cm²). A ramping …

Branches Of Radial Solutions For Semipositone Problems, Alfonso Castro, Sudhasree Gadam, Ratnasingham Shivaji

All HMC Faculty Publications and Research

We consider the radially symmetric solutions to the equation −Δu(x) = λƒ(u(x)) for x ∈ Ω, u(x) = 0 for x ∈ ∂Ω, where Ω denotes the unit ball in R^N (N > 1), centered at the origin and λ > 0. Here ƒ: R→R is assumed to be semipositone (ƒ(0) < 0), monotonically increasing, superlinear with subcritical growth on [0, ∞). We establish the structure of radial solution branches for the above problem. We also prove that if ƒ is convex and ƒ(t)/(tƒ'(t)−ƒ(t)) is a nondecreasing function then for each λ > 0 there exists at most one positive solution u such that (λ, u) belongs to the unbounded branch of positive solutions. Further when ƒ(t) = tp − k, k > 0 and 1 < p < (N + 2)/(N − 2), we prove that the set of positive solutions is connected. Our results are motivated by and extend the developments in [4].